 day. So it's my pleasure to introduce Una Bazou, who is giving a talk on activity-driven transport in harmonic chains and who could not come here because it's still very difficult to travel. And we are very happy that you nevertheless accepted our invitation and we are looking forward to your talk. Thank you, Sarah. Yeah, I'd like to start by thanking the organizers for inviting me in this conference and giving me the opportunity to speak online, even though I could not be there. So I'm going to talk about activity-driven transport in harmonic chains. So this is the sort of outline of this talk. First, I'll very briefly recapitulate the energy transport in a thermally-driven harmonic chain. And then I'll go over to the case when the drive is active, not the usual thermal one. And in particular, I'll focus on two observables. The first one is energy current and the other one is kinetic temperature profile. And then I'll conclude with some open questions. So yeah, energy transport in one dimension. So whenever we talk about energy transport, the typical setup we have in mind is the macroscopic system, which is subjected to temperature gradient and then energy flows from the hot air end to the cooler end of the system. And the simplest theoretical model to understand is the phenomenon is an oscillator chain, in particular in 1D, an oscillator chain which is connected to two thermal reservoirs at the two ends. And the point to remember is that these reservoirs, thermal reservoirs are in equilibrium themselves. So each reservoir is in equilibrium, even though the temperature could be different. So the reservoir itself satisfies fluctuation dissipation theorem, which means that the coupling to the reservoir has to satisfy certain conditions. And the relevant questions that are usually asked in this setup is what is the energy current, what is the thermal conductivity, and how is the local temperature profile and velocity fluctuations, how does it behave locally, et cetera. So the paradigmatic example is the harmonic chain connected to Langeva mass, which was studied by Lijder, Leibowitz, and Lieb in 1967. So let me briefly go over to the model and the results that they obtained exactly. So we are talking about a harmonic chain. So a chain of N harmonically coupled identical, meaning say mass oscillators. And the LF oscillator has a displacement axle from its equilibrium position. And the boundary oscillators, so L equal to 1 and N, they are coupled to two different thermal reservoirs at temperatures T1 and TN. And this coupling is, in the most simplest case, is modeled by adding two forces on the boundary oscillators. One is the dissipation, this gamma x dot kind of term. And the other one is a Gaussian white noise. And because the reservoir is in equilibrium, that means that the strength of this noise is related to the dissipation in a certain way. So we have two different thermal reservoirs at the two ends, with different temperatures T1 and TN, and each of which satisfies fluctuation theory. So the temperature, I mean, the noise at each end is related to the dissipation at that end in a certain way. There are of course various other models of thermal reservoirs. I mean, it's possible to model in various other ways. But this condition of fluctuation dissipation theorem must be satisfied by all. But we'll stick to the most simplest model here. So for this simple model, linearity and Gaussianity allow for an exact solution in the stationary state. And they showed, the RLM paper showed that eventually it reaches a non-equilibrium stationary state, although it's Gaussian and it carries a current. And most importantly, in the thermodynamic limit, that's when the number of oscillators is very large, the current that flows through the system is proportional to the temperature difference of the two reservoirs. It's simply proportional to T1 minus TN. The proportionality constant depends on the mass of oscillators and the constant, I mean, the harmonic constant kappa and all these things. And another important point is that when one looks at the local temperature, it remains uniform at the bulk at the average of the two temperatures at the two ends. So the bulk temperature is simply T1 plus TN by two, and the current flowing is proportional to T1 minus TN. So a natural question to ask is what happens when reservoirs are away from equilibrium themselves? So the reservoirs that we are connecting to the system are not in equilibrium. And in particular, what happens, the question that we are going to ask here is what happens when one considers active reservoirs. So by active reservoirs, I simply mean a medium consisting of active particles. We have already heard a few talks about active particles in this conference. So these are self-propelled particles. Examples are like bacteria, genus particles, and various other kinds of microswimmers and nanoswimmers. They are inherently non-equilibrium in nature. So the barge that the makeup, the reservoir that the makeup would also be out of equilibrium. And recently, there have been many studies, both theoretical and experimental, which study the behavior of single probe particles in such active reservoirs. And they already show various kinds of intriguing features like images of memory, modification of equilibrium theorem and all these things. So what we are going to ask here is what happens? How are the transport properties of extended systems affected when one connects such a system to active reservoirs? So this is like a cartoon. You can think of a system connecting to material bars, just a cartoon representation. And the bars are different. Their activities are different. And we are going to take the most simple model again, just the linear harmonic chain. And we use a very simple model for the reservoir as well. So let me define the model more precisely. So as before, we have a chain of n harmonically coupled oscillators, which we spent XL for the inlet oscillator. And the boundary oscillators, L equal to one and n, are coupled to these active reservoirs now. So how do we model these active reservoirs? We simply add an extra active force, which does not satisfy fluctuation dissipation theorem, and which has a memory. So apart from the usual thermal noises, so the dissipation constant dissipation and white noise at the two ends. So we have xi one and xi n. We have two additional forces, which I call f one and f n, at the two boundary oscillators. These f one and f n are exponentially correlated. So they have memory. So that gives non-marcopian nature of the system. And most importantly, because this is completely independent of the dissipation. And so they do not satisfy any kind of fluctuation dissipation theorem. The reservoirs themselves are away from equilibrium by virtue of this kind of modeling. And this correlation time scales tau one and tau n, they are the measure of reservoir activity. So when tau one and tau n are very small, we go towards the passive limit. And when tau is large, that's more active reservoirs. Okay, so as I already said, so presence of this correlated active force that makes the dynamics, this XB dynamics non-marcopian. But more importantly, because this force is not related to the dissipation, there's no FD in the system. And that's what is going to give rise to certain interesting properties as I'm sure I'll be later. But before that, let me give some examples how one can generate such exponentially correlated forces. So suppose we have, suppose this force FD is simply a run on table like force. So FD is some sigma t, sigma is a dichotomous noise, which flips its sign with some red alpha. In that case tau, the time scale is simply one by two alpha. Similarly, one can think of the boundary particle as an active Brownian particle where FD is cos theta, we also saw this, I mean, each of these examples we have already seen in the in the conference. So cos theta with theta being a Brownian motion. And similarly, one can think of an active one-stranded process at the boundary also. So we'll not consider any specific details, but just the case that the active force has an exponentially correlated form, exponential correlation. And we'll see that that's enough to compute the energy current and temperature profile, which are going to be the quantities of primary interest for this talk. And yeah, so we'll be concentrating in the stationary state. And yeah, because the system is linear. So that again allows exact computation of certain observables, the observables we are particularly interested in. And I'll not go through the details of the computation, but just mentioned that we have used the matrix method introduced by Opishek Thor in this 2001 work. And using that method, one can show that in the stationary state, the displacement of the oscillator can be expressed in terms of a matrix G. This matrix G is the inverse of a tri-diagonal matrix, this here, I hope you can see my cursor. So this G matrix contains the properties of the system and the dissipation gamma. It does not depend on the noise. The noise appears here. So we have zy-tilde, which is the Fourier transform of the white noise, the thermal noise, and f-tilde, which is the Fourier transform of the active. So these two are have very different properties. As far as the white noise goes, the correlation is simply proportional to the temperature. Once again, I mean, remember that this white noise comes from the thermal component of the reservoir. So it satisfies fpt. The active noise, on the other hand, does not satisfy fpt. And it has a Lorentzian spectrum, which is reminiscent of the signature of the memory in the real time. So this zy-tilde, this Lorentzian spectrum of the active reservoir, that's the key quantity here. And that is what controls all the observables that we are going to see. So let's first look at the energy current. So by energy current, I mean the average energy flowing from the reservoir through the system per unique time. And that is, of course, same for all oscillators because there is no source or dissipation in the bulk. And then it's most convenient to write it in terms of the energy coming from the left reservoir to the boundary oscillator, which is here. So this is the velocity of the boundary oscillator, and this is the force that is exerted by the reservoir on the bottom. So it's the total force, including thermal component and the active component. And we want to find the average of this instantaneous current. So as it turns out, this can be computed exactly. And because of the linearity of the system, once again, what happens is this total current actually just separates into two different components, one, the thermal component, the other one is the active component. So the thermal component is exactly same as in the case where there is no active force, basically, the case which I did even with study. This thermal current is simply to the difference of the two reservoirs. The other one, the active current is much more interesting. And that's our main quantity of interest. So it can also be computed exactly. And formally, it's given by some integral. I mean, you don't have to bother about the exact form. But just notice that this g tilde, this reservoir spectrum appears here, g tilde at the left reservoir and right reservoir. Okay, so this Lorentzian spectrum is what controls the behavior. And in the in the end going to infinity limit, the thermodynamic limit, when the chain is very large, it can be computed explicitly. And in that case, the contribution comes from only the system phonon band within the system phonon band, so between omega minus two square k by m and plus two square k by m. That's the characteristic frequency of the of the harmonic chain. And, and this is the form of the active current. Again, you don't have to bother about the details. But note that the active current is some rather the difference of two terms, each coming from one, one reservoir. So let's first try to see how it behaves. This is a plot showing this active current as a function of the activity tau one of the left reservoir, for different values of the activity of the right reservoir. So the symbols show numerical simulations with a particular form of active force. Here we took the run and double kind of form and n equal to 64. And the does the solid lines are simply and going to infinity results, of course, the match. And what more is that we see that there are certain very interesting features which become apparent from this plot. First of all, the current is a non monotonic function of the activity drive or activity of one reservoir. So it first increases and then decreases with the with the maximum at some intermediate activity. So this means that the differential conductivity is negative in certain zone. We'll come back to this issue later. Another point of interest is that the current if you follow one car to see that it crosses zero twice, meaning that the current reverses its direction twice, one at the trivial value when tau one equal to tau n, of course, when there is no activity drive, the current has to be zero. But additionally, at a different value of tau one, if the current also changes sign. So these are two features which were both absent in the family different case, the family different case that kind was simply proportional to T1 minus Tn. So no non monotonicity and no reversal except at the trivial point when the when the temperatures are equal. So I'll just quickly describe these two features in a bit more detail. So I'll say already mentioned so negative differential conductivity means that the the this dj d tau is is negative in certain parameterity. So this negative differential conductivity is sort of a counterintuitive phenomenon. So we are increasing the drive and the current is changing non monotonically. And such this negative differential conductivity is possible only away from equilibrium near equilibrium, it's not possible. And there are certain known examples, but all of them are in nonlinear systems, in particular, in presence of some obstacles or or kinetic constants. So here, what we see is that this active drive somehow leads to this NDC in a in a in a completely linear system. So this, it was kind of surprising result for us. And to understand the physical origin somewhat, we looked at the at the, I mean, at the spectra of the system and the reservoir. So the system frequency spectrum, of course, is is picked. So here in this spot in the left panel, you can see the blue curve is the system's frequency spectrum. It has peaks near the boundary, meaning near the characteristic frequency omega C. On the other hand, the reservoir spectrum is a Lorenzian with peak at omega equal to zero, that's this orange curve. And what happens is that the overlap. Okay, yeah, thanks. And yeah, the overlap actually changes non-mortulnically. So here, if you see in the right panel, so the blue curve is for very small activity. And then the overlap increases when we go to tau equal to 0.5, some intermediate value. And then the overlap decreases again, when we go to a very large activity. So this is what gives rise to this non-mortulny behavior of the current as a function of tau. It's also possible to understand this NDC from the perspective of non equilibrium response theory, but I am not going to the details of that here. Let me quickly talk about the current reversal. So here is a plot of the current in tau and tau n plane. So here the green and the blue shaded regions show where current is positive and negative. So if we vary tau one by keeping tau n fixed, we see what we saw in the previous plot. So first the current is negative, and then it becomes positive, and then it becomes negative again. So here this curve here is a rather non trivial car, which I mean, it's a special car where the current is zero, even though there is a finite activity drive across the, across the system. Okay, except at so this happens at all values of tau one, tau two, tau one, tau n, except at the subtle point where the current is either positive or negative, depending on which way, which parameter we back. So this is also a kind of interesting result, which we don't see in, in, in, when we have thermal or equilibrium reservoirs driving a system, at least a harmonic system. And next I'll very briefly mention what happens to the kinetic temperature profile. So by kinetic temperature, I mean, the average kinetic energy of, of each oscillator. And that also can be computed exactly. So what, what is surprising is that similar to the thermally driven case, here also, the kinetic temperature profile remains flat at the bulk. So here in this plot, you can see, for a fixed value of tau one, the profile as a function of l for different values of tau n. So this orange curve is for tau one equal to tau n, and these two are two other, other different values. So the, it remains flat at the bulk. And one can show that it's actually of the form, it's half of some curly T one plus curly T n, where T n, T one and T n are depending on the tau one and tau n only. So it raises a possibility that whether one can think of an effective temperature picture where this one can associate this curly T one and curly T n as the temperatures of this active universe. So the answer to that is actually no. So even though, I mean, even though the profile is flat, and it has this form, which is kind of reminiscent of the thermal case, there are certain very significant similarities with the thermal case. One is that even when tau one is equal to tau one, tau n, so there is no activity drive at all. In that case, also, if you see this orange curve here, in the thermal case of the profile in that case is completely flat, no boundary layer at all. Here, there's still a natural boundary layer present. And more importantly, if we look at what happens, if we associate this effective temperatures and compute what would have been the thermal current given these temperatures and whether that matches with the activity current that we actually get, these two do not match. So here in this spot, you can see the solid lines are the actual active currents that we compute. The dashed lines are the projected currents if this were taken to be effective temperatures. So these do not match, except here, if you see in this brown curve, which is for tau n equal to point one and also when tau one is very small. So when tau one and tau n are very small, then we see that these two currents are kind of converging with each other. So that tells us that in this passive limit, when tau one and tau n are very small, there is some sort of a thermal picture emerging. And that is also not very surprising, because if we remember our activity was coming from these active forces, which had exponential correlations with time scales tau's. And when tau goes to zero, those exponential correlations actually emulate the kind of a delta function correlation. And in that case, one can associate an effective temperature, which is of this form. And one can show explicitly that to the leading order, the active current in that case actually does match with the thermal current with this effective temperature. Okay, so I'll quickly conclude. So what we have done is study the energy transport of harmonic chain connected to active reservoirs. Active reservoirs we model very simply just by adding a temporarily correlated force, which gives rise to non-marcovian features in the system. But more over, I mean, more important than non-marcovianity is the breaking of fluctuation distribution theorem. So that makes these reservoirs non-equilibrium. And that gives rise to some surprising behavior, like this negative differential conductivity and current reversal in the current. And this I just like to point out that these are very robust results. So we did not use any particular form of active force, we just assumed that it's exponentially correlated. So even if we take an RTP like dynamics or ABP like dynamics or some other dynamics, all of which have this exponential correlation, the result would not change. The result would remain exactly the same. And that's all. I mean, there are many open questions, of course, in particular what happens. So here we have used a very simple model of this active reservoir. But what happens if one takes a more realistic model in the sense that if one considers an assembly of active particles and coupling to them like in the cases where single particle probe were considered. And another thing is that whether this NDC and current reversals survive in the presence of a disorder and what happens when there are the oscillators are not linear, but there is some harm in sitting. So thank you for your attention. Thank you very much, Una, for this very interesting and clear talk. So we have time for questions. And I also want to encourage people in the chat. And of course, also for the people in the audience, we'll give priority to students first. But we already have a question in the chat, so I will just read it out. So in general, when the system and the reservoirs are strongly coupled, is it easy to identify what the energy current between the system and the reservoir is? For strong coupling, I would expect also a finite contribution from the coupling Hamiltonian. So here there is no coupling. Which coupling Hamiltonian do you mean? The coupling with the reservoir is not happening to a Hamiltonian. It's just dissipation and noise. So maybe I did not understand the question. Can you clarify which coupling Hamiltonian you mean? So Tobias Becker, I hope you can hear her. If you would like, you can type something in the chat. Okay, okay. Yeah, and there's a student first. Okay, I'm going to release it. Just out of curiosity in this phase diagram, where you show the sort of shrinking of this reverse current phase. Exactly. There's like that point where the two reversal regions meet. Yeah, does it have some interesting features? So this is just a subtle point in that. So if you see this, I did not show this. So this is the plot of the current. I mean, the 3D plot. So this crossing of the two cars, that's a subtle point. And so if tau n is equal to this tau bar and we change tau one, then the current remains negative all through. Okay, except it touches zero at this point. So there is no, there is no reversal, it remains negative and touching this zero at this point. And on the other hand, if we fix tau one at this value and change tau n, then the current remains positive throughout. That's the feature of this point. Ayurna. So a question. So do you expect anything interesting in the large deviation function of the current? Possibly, we have not looked at the large deviation yet. But possibly, yes. So here, of course, I mean, once we have this active drive, the whole system is not doubted anymore. And we expect some signature of that also appearing in the large deviation function. Okay, we have another question. Hi, it's Marco. Hi, hi. In terms of magnitude, is it as can it become as large as the thermal current or it's of another order of magnitude? Which one? The current itself? I mean, this active current, is it as big as the other one? Or it's much smaller than the thermal current? Yes, I think it, yes, I think it can be. I mean, there is no, there is no relative competition between these two. So the actual magnitude will depend also on the strength of the noise. So here this is, so the correlation that we had, if FTFT prime, that also has a strength. And depending on if we increase that strength, the current would also, I mean, the overall magnitude can be scaled. I also have a question, Oana. So I was interested in the temperature profiles that you showed. And you showed that's different from the case of just thermal drive, your boundary effect spreads a little bit into the bulk, right? So there is a little bit of spreading into the bulk. And I was wondering if you know if that is because of the activity or is it because of the memory? In other words, if the two bars would have memory but would be FTFT fulfilling, would you also have this spreading or not? Do you know that? I don't, but I can check. I think it's very interesting because, of course, for equilibrium systems. This is exponential. I mean, the boundary layers are exponential, that we did check. Whether it survives, if one does not have activity but still non-Marcovian. Yes, I don't know. Okay. Yeah, very interesting. Thank you. Okay, I think we had a lot of discussion. And if there are more questions, you can send them to Oana in the chat. And I think we should, or to me, and we can move on to the next speaker and thank Oana again very much for the nice talk.